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ON  THE  IMPRIMITIVE  SUBSTITUTION 
GROUPS  OF  DEGREE  PIFTEEII  AM)  THE 
PRIMITIVE  SUBSTITUTION  GROUPS  OP 
DEGREE  EIGHTEEN 

by 
Emilie  Norton  Martin 


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'  ON   THE   IMPRIMITIVE  SUBSTITUTION  GROUPS 
OF  DEGREE  FIFTEEN  AND  THE  PRIMITIVE 
SUBSTITUTION  GROUPS  OF  DEGREE 

EIGHTEEN 


A    DISSERTATION 

PRESENTED  TO  THE  FACULTY  OF.BRYN  MAWR  COLLEGE  FOR  THE  DEGREE  OF 

DOCTOR  OF   PHILOSOPHY 


Bv  EMILIE    NORTON   MARTIN 


iqoi 
Z^  £orb  (§attimott  (prcee 

The  Friedenwald  Company 
baltimore,  mu.,  u.  s.  a. 


ON  THE   IMPRIMITIVE  SUBSTITUTION  GROUPS 

OF  DEGREE  FIFTEEN  AND  THE  PRIMITIVE 

SUBSTITUTION  GROUPS  OF  DEGREE 

EIGHTEEN 


A   DISSERTATION 

PRESENTED  TO  THE  FACULTY  OF  BRYN  MAWR  COLLEGE  FOR  THE  DEGREE  OF 

DOCTOR  OF  PHILOSOPHY 


By  EMILIE   NORTON   MARTIN 


IQOI 

The  Friedenwald  Company 
baltimore,  md.,  u.  s.  a. 


Engineering  & 
Mathematical 

Sciences 

Library 


MS4><5-' 


Oil  the  Imprimitive  Substitution  Groups  of  Degree 
Fifteen  and  the  Primitive  Substitution 
Groups  of  Degree  Eighteen. 

By  Emilie  Norton  Martin. 


The  following  work  is,  with  some  slight  modifications,  the  same  as  that  of 
which  an  abstract  was  presented  at  the  summer  meeting  of  the  American  Mathe- 
matical Society  in  1899.  With  regard  to  the  imprimitive  groups  of  degree  fifteen, 
which  form  the  subject  matter  of  the  first  part  of  this  paper,  it  should  be  stated 
that  I  have  added  two  new  groups  to  the  list  as  originally  presented,  namely,  the 
groups  with  five  systems  of  imprimitivity  simply  isomorphic  to  the  alternating 
and  symmetric  groups  of  degree  5,  and  that  Dr.  Kuhn  reported  at  the  February 
meeting  of  the  Society,  1900,  that  he  had  carried  the  investigation  further,  adding 
28  to  the  70  groups  that  I  succeeded  in  finding. 

In  the  second  part  of  this  paper  the  determination  of  the  primitive  groups 
of  degree  18  depends  to  a  great  extent  upon  the  lists  of  transitive  groups 
of  lower  degrees  already  determined.  Any  new  discovery  of  groups  of  degree 
less  than  18  would  necessitate  an  examination  of  such  groups  to  determine 
whether  they  can  be  combined  with  others  in  such  a  way  as  to  generate  a 
primitive  group  of  degree  18.  This  list,  therefore,  cannot  claim  to  be  abso- 
lutely complete,  since  omissions  are  always  possible. 

Imprimitive  Svhstitution   Groups  of  Degree  Fifteen. 

Every  imprimitive  group  contains  a  self-conjugate  intransitive  subgroup 
consisting  of  all  the  operations  that  interchange  the  elements  of  the  systems  of 
imprimitivity  among  themselves  without  interchanging  the  systems.  Therefore, 
the  problem  of  the  determination  of  all  imprimitive  groups  of  degree  15  falls 
into  two  parts:   1st,  the  determination  of  all  intransitive  groups  of  degree  15 

21643.3 


2  Martin  :   On  the  Imprimitlve  Substitution  Groups  of  Degree 

capable  of  becoming  the  self-conjugate  subgroups  of  such  imprimitlve  groups ; 
2d,  the  determination  of  substitutions  that  will  interchange  the  systems  of 
imprimitivity  and  at  the  same  time  fulfill  other  conditions  depending  upon  the 
particular  group  under  discussion.  The  intransitive  self-conjugate  subgroup  is 
called  for  shortness  the  head,  the  remaining  substitutions  of  the  imprimitive  group 
are  designated  as  the  tail,  a  terminology  that  has  been  adopted  by  Dr.  G.  A. 
Miller  in  his  papers  on  imprimitive  groups. 

The  elements  of  an  imprimitive  group  of  degree  15  may  fall  into  three  sys- 
tems of  five  elements  each,  or  into  five  systems  of  three  each.  For  the  first  of 
these  cases,  certain  theorems  given  by  Dr.  G,  A.  Miller  (Quar.  Jour.  Math.,  vol. 
XXVIII,  1896)  are  useful.  With  a  slight  modification  in  notation  in  order  to 
adapt  them  to  the  notation  of  this  paper,  they  are  as  follows,  where  G^  repre- 
sents a  group  in  the  elements  with  index  1,  while  G^  and  G^  represent  precisely 
the  same  group  in  the  elements  with  indices  2  and  3. 

Theorem  I. — All  the  substitutions  that  can  be  used  to  construct  tails  are 

{a\al  ....  a\)  all  {ajal  .  .  .  .  a^)  all  {alal  ....  al)  all 

\  (a\alal .  alajal aWnCil) .  («i«i  •  «2«2 «n«n)  f 

—  {a\al  .  .  .  .  al)  all  (afal  ....  a^)  all  {alal  ....  a')  all. 

Theorem  II. — If  G^  =  ((A^l  •  •  •  •  «n)  all,  there  are  three  imprimitive  groups 
with  the  common  head  {G^G~G^)  pos,  and  two  with  the  common  head  G^  pos  GF  pos 
G^  pos  -+-  G^  neg  G^  neg  G^  neg. 

Theorem  III. — If  G^  =.  (ala]  ....  a\)  pos,  there  are  three  imprimitive  groups 
luith  the  common  head  G^G~G^,  and  three  with  the  common  head  {G^6r^G^)i^  j^  i . 

Theorem  IV. — If  the  head  w  G^G^G^,  there  is  only  one  group  which  corresponds 
to  (abc)  eye. 

The  possible  heads  for  these  groups  are  got  either  by  the  direct  multiplica- 
tion of  transitive  groups  of  degree  5  in  the  three  systems  of  elements,  or  by  the 
establishment  of  isomorphic  relations  between  such  groups. 

The  transitive  groups  of  degree  5  are  five  in  number,  and  fall  naturally  into 
two  categories,  the  first  containing  the  symmetric  group  and  its  self-conjugate 
subgroup,  the  alternating  group,  the  second  containing  the  metacyclic  group, 
together  with  its  two  self-conjugate  transitive  subgroups.  These  five  groups  are 
represented  respectively  by 

(aia^asa^a^)  all,  {aia^a^a^a^)  pos,  ((ha^a^a^a^).;^,  {a^a^asa^a^^o,  {aia^asa^a^)^ . 


Fifteen  and  the  Primitive  Substitution   Groups  of  Degree  Eighteen.         3 

P>om  the  first  two  groups  come  the  following  heads : 

I.        {a\alalalal)  all  (alalaldlal)  all  {alalalalal)  all  =  A7280oo- 
11.      \(a\alalalal)  all  (alalalalal)  all  (afa^c4«4«l)  all|  pos  =  Esmm- 

II I .  (alalalalal)  pos  (a?a|a|a|a^)  pos  (af<i|a^a^a|)  pos 

+  (a}c4r4«l«^5)neg(aja|«^a^a^)neg(afa|a|a|«i)  neg  =  ^/432ooo- 

IV.  {a\cdala\al)  pos  (a]a^a|a|al)  pos  {alcdaldldl)  pos  =  //gieooo- 
V.       {a\alalalal .  alalalalal .  alalajalal)  all  =  Hi2o- 

VI.        (alalalalal.  dfdlalalal.  ajalalalaf) -pos  =  H^q. 

From  the  three  remaining  groups  come  the  heads  : 

VII.        (alalalalaiy^o  («i«If'l«4^5)2o  («i«i«3«'4«5)3o  =  ^sooo- 

VIII.  {{alalalalaDzo  («?«!' 3^' 4« 5)30  («i«2«!«4«5)3o f  pos  =  i?4ooo- 

IX.  J  (alalalalal) ^f,  («^''>«|al«I)2o  («i«3«Ift!(''5)2o }  10, 10, 10  =  ^sooo- 
X.        (alalalalaWo  («i«a«3«4«5)io  («i«l«3(«4«5)io  =  ^looo- 

XI.  -I  (alala]alal)oo  (alalalalaDsa  (oi^i^l^^^D^ol  5, 5, 5  =  ^soo- 

XII.  ]  (alalqlalaDiQ  (aia|«^a^a|)io  (alahlalafji^i  \  5, 5, 5  =  ^250- 

XIII.  (alalalalal)  eye  (alafalalal)  eye  (ajalaga^al)  eye  =  H^z^. 

XIV.  (aja2<^3c4«5  •  (^i(«|«3«l«5 .  ai«|«|«|«|)oo  =  -Sao- 
XV.        (alalalalal .  a^ala^alal .  «i«3<^<3'/4«5)io  =  -^lo- 

XVI.        (alalalalal .  afalalapl .  alalalaldf)  eye.  =  H^. 

The  groups  corresponding  to  these  heads  may  be  isomorphic  either  to  (a^d\i^) 
eye  or  to  (a^a^a^)  all.  To  generate  a  group  isomorphic  to  (a^a^a^)  eye  a  substitu- 
tion with  the  following  properties  must  be  added  to  the  head :  it  must  have  its 
cube  in  the  head,  it  must  interchange  all  three  systems,  and  it  must  transform 
the  head  into  itself.  Calling  the  group  so  found  G,  the  groups  isomorphic  to 
(a^a^a^)  all  may  be  found  by  combining  with  G  any  substitution  that  has  its 
square  in  the  head,  that  interchanges  two  of  the  systems  leaving  the  third 
unaffected,  and  that  transforms  H  into  itself,  and  G  into  itself. 

As  all  the  heads  given  above  are  symmetric  in  the  three  sets  of  elements,  each 
head  furnishes  two  groups  by  means  of  the  symmetrically  formed  substitutions 

s  =  a{aiai .  alar,al .  a^ayil .  a\aiai .  atfliai ,   t  =  a\ai .  a\a% .  a^a^ .  aial .  a'^a^  . 

The  letters  s  and  t  are  used  throughout  this  section  of  the  paper  to  denote  these 
particular  substitutions,  other  substitutions  fulfilling  the  same  conditions  being 
denoted  by  the  same  letters  with  suflQxes. 


4  Martin  :    On  the  Imprimitive  Substitution   Groups  of  Degree 

According  to  Theorem  I,  anys„  or  ^„  must  be  the  product  of  some  substitution, 
(7„,  of  the  most  general  head,  ^17289001  by  s  or  t.  Therefore  a^  must  be  a  substitution 
of  a  subgroup  of  ^j^ogooo  that  contains  the  special  5"  under  consideration  as  a  self- 
conjugate  subgroup. 

We  may  now  proceed  to  the  determination  of  the  groups  to  be  derived  from 
the  various  heads  taken  in  order. 

I.  ^1728000  gives  us,  according  to  Theorem  I,  only  the  two  groups, 

j^i728ooo,  s,  i  of  order  5184000i, 
and   J5i728ooo,  «,  t\  of  order  10368000. 

II.  j5864ooo  gives  us,  in  accordance  with  Theorem  II,  three  distinct  groups. 
Of  these,  two  are  the  groups, 

j^eeiooo,  A  of  order  2592000i, 
)  5864000.  ^^  i\  of  order  51840002, 

A  a  that  transforms  the  head  into  itself  without  belonging  in  the  head  is 
a  =:  a\a\ .  This  cannot  be  combined  with  s ,  as  {psY  is  an  odd  substitution  ;  it  may, 
however,  be  combined  with  t .     The  remaining  group  is  therefore 

•|  i?864ooo.  '5,  «i«2 .  ^[  of  order  5I840OO3 . 

Of  these  two  groups  of  order  5184000,  the  first  contains  both  odd  and  even 
substitutions,  the  second  only  even. 

III.  jS432ooo  gives,  by  Theorem  II,  the  two  groups 

]^4320oo>«t  of  order  1296000i, 
1^432000,  s,t\  of  order  25920000. 

IV.  5216000  gives  us,  by  Theorem  III,  three  distinct  groups,  a  =  a\a\  trans- 
forms the  head  into  itself,  but  when  combined  with  s  it  gives  an  odd  substitu- 
tion whose  cube  cannot  be  found  in  the  head.  The  substitution  at  furnishes  us 
however,  with  a  new  t^.     The  three  groups  are,  therefore, 

l^oieooo,  s\  of  order  648000, 
\Rmm^  s,t\  of  order  1296OOO2, 
1^216000.  «.  «i«2-^l  of  order  1296OOO3. 

The  two  groups  of  order  1296000  are  distinct,  since  the  one  contains  both  odd 
and  even  substitutions,  the  other  only  even. 


Fifteen  and  the  Primitive  /Substitution  Groups  of  Degree  Eighteen.  5 

V.  ff^oQ  is  not  contained  self-conjugately  in  any  larger  subgroup  of  5"i728ooo> 
therefore  only  the  two  following  groups  can  be  formed  from  it : 

{^120,  s\  of  order  360i, 
\Hi2o,s,  t\  of  order  720. 

VI.  ^eo  gives,  in  accordance  with  Theorem  TIT,  three  groups  : 

J^eo,  s]  of  order  180, 

\ITqo,  s,  t\  of  order  3602, 

\  H^Q,  s,  a\al .  «!«! .  a^al .  ^[  of  order  BGOg. 

The  last  of  these  groups  consists  entirely  of  even  substitutions. 

The  remaining  heads  are  all  composed  of  substitutions  of  the  type 

VliW^^V^a^U^vl.ut,  (1) 

where  v,,i^=-  aldlala],  u^i=  a\alala\al,  while  v^i,  Ua2,  v^^,  w^s  denote  the  same  sub- 
stitutions written  in  elements  with  the  indices  2  and  3  respectively.  The  substi- 
tutions Va'y  Ua^  generate  the  metacyclic  group  in  the  five  elements  with  index  1, 
these  substitutions  being  subject  to  the  conditions 

The  most  general  s^  is  given  by 

From  this  we  find 

,9f  =  <+^  +  'n,^.<  +  *  +  ''7^S.<3+''  +  '^iC,  (3) 

where  X=2'{  2^  -f  Z^i)  -f  Cj ,  ^ 

v=  2'^(2''ci  +  ai)  -f  ^.  J 

Transformation  of  the  general  substitution  (1)  by  s^  gives  us 

Sa^K^  u-a^  v^^,  iC  vl^  ut  s^  =  vl  ulx  vl,  ul.  v^^.u''^,,  ( 5 ) 

where  ;^  =  «!  -f  2"a'  —  Ta-^ ,  -j 

l,  =  h,  +  2'^'-2%,^  (6) 

1.  =  Ci  +  2 Y'  —  2^Ci ,  ) 

The  general  substitution  of  the  group  G=  \TI,  s^\  is 

T=slv:.u^:.vl.u^vl.itt..  (7) 


6  Martin  :    On  the  Imprimitive  Substitution   Groups  of  Degree 

The  most  general  t^  is  given  by 

t^  =  vluOviluilv%u'^,t.  (8) 

Upon  squaring  this  substitution,  we  get 

tl  =  v'',^  + "'  M^  vll  +  '^  K.  v'J^  K,,  (9) 

where  2,=  ^^'Gs  +  bs,^ 

fi=2''^b,+a,l  (10) 

On  transforming  the  general    substitution  T  by  the   general  t^,   we  have,  after  a 
straight-forward  calculation,  the  following  expression  for  the  case  x=  I: 

tj'T^^,t^  =  s^^v-.''^  +  '''  +  ^+''K.vl\-'^-^''  +  UCv-.'^  +  '^  +  ''  +  ''ii':,^,  (11) 

where  ^=  —  2''^-"»  +  ^+V3  +  2'  +  '^ai  +  2*^/3'  +  h,,  ^ 

r  =  —  2'^'-*'  +  ^  +  ''Z>3  +  2^+'^^&i   +  2'y'  +  Cg.  j 

We  may  now  return  to  the  consideration  of  special  groups. 
VII.     i/gooo  gives  only  the  two  groups  formed  with  s  and  t,  as  any  a  that 
might  be  used  is  already  contained  in  this  head.     The  groups  are,  therefore, 

{jSgooo,  s\  of  order  24000i, 
^^eooo,  s,  t\  of  order  48000. 

YIII.  iTiooo  ^^s  the  general  substitution  (1)  subject  to  the  condition  a -\- (3 
+  y=  0  (mod  2).  From  (3),  it  is  evident  that  s„  is  subject  to  the  condition  a-{-b 
-}-  c  =  0  (mod  2).  Therefore,  s^  is  already  in  the  group  generated  by  ^4000  and 
by  s,  and  there  is  only  one  group  isomorphic  to  (a^a^a^)  eye.  We  find  by  (9) 
that  every  t^  has  its  square  in  the  head,  and  by  (11),  that  every  f^  transforms 
the  head  into  itself,  therefore,  we  may  take  as  a  new  t^  the  simplest  substitution 
for  which  ^3  -\-  b^  +  Co=  1  (mod  2),  viz.  : 

Vat  =  a\a{ .  cqa^a^a^rr^aia^iao. 
The  three  groups  with  this  head  are,  therefore, 

]^40oo>  -^f  of  order  I2OOO1, 
jFjooo.  5.  ^f  of  order  24OOO2, 
J-&4000'  *'  ^a^|-  of  order  24OOO3. 


Fifteen  and  the  Primitive  Substitution   Groups  of  Degree  Eighteen.         7 

Of  these  groups  the  first  and  third  consist  of  even  substitutions,  the  second  of 
even  and  odd. 

IX.  ^2000  has  the  general  substitution  subject  to  the  condition  a  =  ^  =  y  (mod 
2).  From  (3)  and  (5),  it  is  phxin  that  every  s^  can  be  used  to  generate  a  group  of  the 
kind  required.  The  only  possible  form  for  the  cofactor  of  s,  if  it  is  not  to  give 
the  group  generated  by  s  and  the  head,  is  Vaivl-iV^i,  where  a,  b,  c  do  not  fulfill  the 
condition  a  =  b  =  c  (mod  2).  The  simplest  form  for  such  a  cofactor,  and  a  form 
to  which  all  others  reduce,  is  found  by  making  two  of  the  exponents  vanish  and 
the  third  become  equal  to  1,  e.  g.,  Si==  VgiS  ==  a\(ila\  .alalalalalalaldlalalalal. 
Now,  81=:  Si  .VaxVa^Vas  aud  sl  =z  sf .  vliV^ivls  ]  we  may,  therefore,  take  s\  as  the  s. 
in  the  place  of  §1  and  still  have  the  same  group.  But  si^=  (vl-^vlz)'^  s  {vliV%)^ 
therefore  the  group  we  have  now  found  is  merely  the  transformed  of  the  group 
generated  by  s  with  respect  to  the  substitution  v^vlz.  Consequently,  there  is 
but  one  group  corresponding  to  the  cyclic  group  of  degree  three. 

If,  in  addition  to  the  group  given  by  t,  we  have  a  group  given  by  t^,  then 
according  to  the  relations  derived  from  (11),  a.^^b.y^c^  (mod  2),  i.  e.,  the  pos- 
sible values  of  ^^  are  already  present  in  the  group  generated  with  the  help  of  t. 
The  two  imprimitive  groups  with  this  head  are,  therefore,  the  groups 

]^2ooo.  s\  of  order  6000. 
j/^Mooo,  s,t\  of  order  1200O2. 

In  this,  and  all  following  work,  the  terms  u  in  the  cofactors  of  s  and  t  are 
taken  as  unity,  unless  the  contrary  is  expressly  stated. 

X.  ^jooo  has  its  general  substitution  subject  to  the  condition  a  =  l3  =  y  =  0 
(mod  2).  By  Theorem  IV,  this  head  gives  only  one  group  isomorphic  to 
{abc)  eye.  If,  in  addition  to  the  substitution  t,  there  is  a  substitution  fp,  the 
relations  satisfied  by  the  exponents  of  the  v^s  in  (11)  reduces  to  a^^ib-i  —  c^ 
(mod  2).  We  have,  therefore,  two  distinct  groups  according  as  ag  is  even  or  odd. 
The  three  groups  with  this  head  are 

j  ^1000 '  * )  of  oi"der  SOOOj , 
j^iooo.  s,  t]  of  order  6OOO2, 
]^iooo>  '5,  Va^Va^VaJ]  of  Order  6OOO3. 

XI.  ^500  subjects  the  general  substitution  to  the  conditions  a=^  ^  =■  y,  where 
a  =:  0,  1,  2,  3.  Since  every  substitution  s^  satisfies  the  necessary  conditions,  the 
following  independent  types  of  s„  must  be  examined:  Va^s,  vl.s,  vliS,  Va^vl^s.     The 


8  Martin  :    On  the  Iinprimitivc  Substitution  Groups  of  Degree 

fourth  power  of  these  substitutions  is  in  every  case  the  transformed  of  s  with 

respect  to  some  combination  of  the  ^-'s;  therefore,  they  give  nothing  new.     The 

possible  forms  for  t^  are  derived  from  the  equation  easily  deducible  from  (11); 

—  Co  +  b^^Oi  —  c.^  =  —  ^2  +  ^2  (mod  4),  which,  taken  in  conjunction  with  the 

limited  range  of  values  of  ao'  ^2-  ^z^  gives  org  =:  Z>2  =  c^.     That  is,  every  possible  t^ 

is  alread}^  included  in  the  group  generated  by  t.     This  head  gives  accordingly 

only  the  two  groups, 

]^5oo,  s\  of  order  1500i, 

^iJgoo,  s,t\  of  order  SOOOg. 

XII.  i?25o  subjects  the  general  substitution  (1)  to  the  conditions  a  ==  /?  =  }/=0 
(mod  2).  To  determine  an  s^,  we  have  from  (3)  the  condition  a  +  6  +  c  =  0 
(mod  2).  An  examination  of  the  four  apparently  distinct  types  of  s„,  v\s,  v^v.^s, 
vfv^s,  vlvls,  shows  that  just  as  in  the  last  set  of  groups,  these  each  give  a  group 
that  can  be  derived  from  the  group  generated  by  s  by  means  of  an  easy  trans- 
formation. 

The  possible  forms  f^  must  fulfill  the  conditions,  deducible  from  (1 1),  — «  g  + 
b2  =  —  60  +  02  =  —  C2-{-  a2  =  0  (mod  2)  and  also — 02  +  62  =  ^2  —  ^i  (mod  4). 
These  reduce  to  the  simple  condition  a^  =  bo  =  Co,  which  furnishes  the  substi- 
tution t^  =  Va^  Va7  v^z  t.     Thls  hcad  gives  therefore  the  three  groups, 

]^2oo»  ^1  of  order  750i, 
{Eo^Q,  s,  t\  of  order  150O2, 
]J525o,s,  Va^Va-iVai  t\  of  ordcr  ISOOg. 
The  second  group  alone  contains  odd  substitutions. 

XIII.  ^125  gives  in  accordance  with  Theorem  IV  only  one  group  in  which 
the  systems  are  interchanged  cyclically.  The  general  substitution  of  this  head  is 
subject  to  the  condition  a  =  /?  =  ^  =  0.  Applying  this  condition  to  (9)  and  (11) 
we  find  (^2  ^=  ^2  =^  <^2 '  while  a^  lies  under  the  further  restriction  of  being  even. 
Therefore  we  have  in  addition  to  t  the  substitution, 

J      ">        2        5      J   1"  12  1''  A       '>  12  ^3  S3 

i^  =.  Va^v^^K^  t  =  a\ai.  aias.agaj.ojag.  aiai.  aoal.alal. 
The  three  groups  given  by  this  head  are, 

1^125,  s\  of  order  375, 

\Ei2bi  ^)  i\  of  order  75O2, 

{^i25»  s,VaiVa2Va3t\  of  ordcr  750g. 

XIV.  ^20  imposes  upon  the  exponents  of  the  general  term  the  conditions 


Fifteen  and  the  Primitive  Suhstitation  Groups  of  Degree  Eighteen.         9 

a  =  13  =  '/,  a'=-P'=.y'.  Making  use  of  this  in  (5)  and  (6)  we  find  2"a'  = 
2V  =  2'^  a'  (mod  5) ,  which  gives  at  once  az=.h=ic.  Using  this  latter  equaUty  in 
the  equations  that  are  deduced  from  (3)  and  (4)  we  find  a^  =  5^  =  Ci  with  the 
single  exception  of  the  case  a  =  0,  where  the  equations  become  indeterminate, 
being  satisfied  by  every  value  of  a^,  Sj,  c^.  An  examination  of  all  of  the  appa- 
rently independent  sets  of  value  for  %,  6^,  c^  shows  that  in  every  case  the  group 
is  transformable  into  that  generated  by  .9  alone.  In  order  to  determine  all  sub- 
stitutions t^  we  use  the  equation,  derived  from  (11),  — a^_  +  So^ag  —  ^3  =  '^2  —  ^3 
(mod.  4),  from  which  follows  at  once  t/g  =  b^  =  c.,.  From  (12),  by  making  use  of 
the  special  case  a  =  (3  =^'y  =^  0,  can  be  derived  the  relations  —  a^-\-  1)^  =  —  Cg 
+  «3  =  —  ^  +  C3  (mod.  5);  i,  e,  ag  =  ^3  =  Cg.  The  only  groups  with  this  head  are 
therefore  the  two  groups, 

]5oo,  s\  of  order  6O1, 

1 530,  s,  t\  of  order  120. 

XV.  HiQ  has  the  general  term  (1)  subject  to  the  conditions  a  =  ;5  =  )/  =  0 
(mod  2),  a'  =  /?'  =  y'.  By  precisely  the  same  line  of  argument  as  that  laid  down 
in  the  preceding  case  we  arrive  at  the  conclusion  a=ib  =  c,  a^  =  b^z=c^,  ^3  = 
S3  =  C2,  ag  =:  &g  =:  Cg.  In  this  work,  too,  the  indeterminate  values  of  a^,  b^,  c^ 
require  a  careful  examination  that  leads  to  no  new  group.  From  this  head  come, 
therefore,  the  three  groups, 

\HiQ,  s\  of  order  30i, 

\HiQ,  s,  t\  of  order  GO,, 

jiTio,  ^.  v„^Va,  Va^  t\  of  ordcr  6O3. 

Of  these  three  groups  the  second  alone  involves  odd  substitutions. 

XYI.  H^  imposes  upon  the  general  term  the  conditions  a=:  (3  =  y  =:  0,  a'  = 
/?'  =  y,  By  arguments  similar  to  those  used  in  the  last  two  cases,  with  the 
further  addition  of  the  condition  imposed  by  (3),  a  +  ^  +  c  =  0  (mod  4),  we  find 
a  =  6  =  c  =  0,  «!  =  &i  =  Cj.  In  the  determination  of  t^  we  see  at  once  from  (9) 
that  C3  must  be  even,  while  from  (ll)  we  find  a3=:Z>3=C3,  and  from  (12) 
a3=bs  =  Cs. 

The  groups  given  by  this  head  are  as  follows: 

j^g,  s\  of  order  15, 

jiTg,  s,  t\  of  order  3O2, 

]  Ilr,,  s,  vl^  v%  vis  t\  of  order  3O3. 


10  Martin  :   On  the  Lmprimitive  Siihstitution   Groups  of  Degree 

Passing  now  to  the  case  of  five  systems  of  three  elements  each,  there  are 
seven  heads  considered  in  this  paper,  six  involving  all  the  systems  symmetrically, 
the  remaining  head  being  unity. 

I.     {a\a\a\)  all  (ala|a|)  all  {a\alaf)  all  {a\a\af)  all  {a\ala\)  all  =  Hm^, 

II.         \n^m\    POS=   ^3888, 
ill.         |i27776j    3,3,3,3,3  ■—  -"486  > 

IV.  {a\a\a\)  pos  {a\a^xv^^  pos {a\alaf)  pos  {a\a\a\)  pos  (aicisal)  pos  =  ^^243 . 

V.  {a\a\a\  .  a\a\a\  .  alalaj  .  a\ala\ .  a\ala\)  all  =  H^ , 

VI.  {a\a\a\ .  alaldi .  alalaj .  a{alal .  alalal)  eye  =  II3 , 

VII.  Unity. 

Denoting  the  system  with  index  r  by  A,,  it  is  evident  that  these  systems 
must  be  interchanged  according  to  the  five  groups  (^.1^2^.3^14^.5)  eye,  (^.1^.3^.3^.4^.5)10, 
(^i^^3^4^)2o,  (A^A^A^A^A^)  pos,  (A^A^A^A^A^)  all. 

The  order  of  procedure  in  each  case   is  as  follows : 

1.  If  the  group  is  to  correspond  to  (^i^2^3-^4^5)  ^y^^  -^  substitution  s  must 
be  found  that  will  interchange  the  systems  cyclically,  transform  the  head  into 
itself,  and  have  its  fifth  power  in  the  head.  The  imprimitive  group  so  generated 
may  be  called  G. 

2.  If  the  group  is  to  correspond  to  (J.i^2A^4^5)io.  it  must  contain  (tj  as  a 
self-conjugate  subgroup.  In  addition,  therefore,  to  the  s  of  case  1,  a  substitution 
t  must  be  found  that  will  interchange  four  of  the  systems  in  two  pairs,  as 
AzA^ .  A^Ai,  while  leaving  the  remaining  system  unaltered,  and  that  will,  at  the 
same  time,  transform  the  head  into  itself  and  G  into  itself.  This  substitution  t 
must  also  have  its  square  in  the  head.  This  imprimitive  group  shall  be 
called  G^. 

3.  If  the  group  is  to  correspond  to  (A^  A.  A3  A^  A^\o ,  it  must  contain  both  Gi 
and  6r3  as  self-conjugate  subgroups.  In  addition,  therefore,  to  the  s  of  case  1,  a 
substitution  u  must  be  found  interchanging  four  of  the  systems  cyclically,  accord- 
ing to  A2A3A^Ai  for  instance,  transforming  Gi  and  G2  into  themselves  and  hav- 
ing its  fourth  power  in  the  head. 

4.  If  the  group  is  to  correspond  to  (A^AoA^A^A^)  pos,  two  substitutions, 
V  and  v',   must  be   found    corresponding    to  ^,  A  Jg  and  Ji  J^iA  •     These  sub- 


Fifteen  and  the  Primitive  Suhstitiition   Groups  of  Degree  Eighteen.       11 

stitutioiis  must,  therefore,  each  interchange  three  systems,  leaving  two  mialtered, 
they  must  have  their  cubes  in  the  head,  and  must  transform  the  head  into  itself. 
This  group  may  be  called  &. 

5.  If  the  group  is  to  correspond  to  [A^A-.A^A^Ar^  all,  two  substitutions,  w 
and  w',  must  be  found  corresponding  to  ^^^-2^-3^4  and  A^A^.  G'  is  to  be  con- 
tained in  this  new  group  as  a  self-conjugate  subgroup,  therefore  w  and  to'  must 
transform  the  head  into  itself  and  G'  into  itself.  The  fourth  power  of  w  and  the 
square  of  to'  must  both  be  contained  in  the  head. 

I.  Hti^q  is  the  largest  possible  intransitive  group  with  the  given  systems  of 
intransitivity,  and,  consequently,  only  one  group  with  this  head  corresponds  to 
each  of  the  transitive  groups  of  degree  5.  For  each  of  these  groups  a  substitu- 
tion or  pair  of  substitutions  can  be  found  fulfilling  all  required  conditions  and 
involving  the  elements  of  the  systems  symmetrically.  A  second  set  could  be 
found  only  by  multiplying  this  first  step  by  some  substitution  belonging  to  the 
largest  group  that  contains  the  head  self-conjugately  without  interchanging  any 
of  the  systems.  But  this  group  is  the  head  itself.  The  required  groups  are, 
therefore,  the  following : 

{57776,  6-}  of  order  38880, 
1^7776,  s ,  t\  of  order  77760, 
] 5777s,  s,  u\  of  order  155520, 
J57776,  v,v'\  of  order  466560i, 
J//7776,  w,  10' \  of  order  933120, 

where  s  =  alalala\al .  alajalalal .  alalalalal, 

.     OS  Ql  25  3     4  an  3     4 

t  =  a-{a\ .  ayq  .  a^^^l .  cdat .  a^cq  .  a^cq , 

o     Q     r     A         a;-!")!         2     3     54 

u  ^=.  aiaiciial .  a2ar,alal .  aicqcqal , 

193  l^q  I"!? 

V  =  a{a^ai .  aUir/i^ .  a^a^ct^ , 
v'  =■  a\alai .  alcdal .  ala'^al , 
w  =  alalala\ .  alalalal .  alalalal , 
2o'  =:a\al.alal.alal. 

These  letters  shall  be  kept  throughout  this  section  of  the  paper  to  denote  these 
symmetrically  formed  substitutions,  other  substitutions  with  corresponding  prop- 
erties being  denoted  by  the  same  letters  with  suffixes. 


12  Martin  :    On  the  Imprimitive  Suhstitution   Groups  of  Degree 

11.  //gggg  gives  only  one  group  isomorphic  to  {A^A.A^A^A-^  gjq,  viz.,  the 
group  generated  by  s.  Any  new  s^  must  have  as  cofactor  an  odd  substitution 
belonging  to  H^^-^,  but  the  fifth  power  of  such  a  substitution  is  not  contained  in 
the  head.  There  are,  however,  two  groups  isomorphic  to  (^1^.2^3^4^5)10,  since 
both  t  and  t„  =  a\al .  t  fulfill  the  necessary  conditions.  The  former  generates  a 
group  (rggggo  containing  only  even  substitutions,  the  latter  generates  a  group 
^38880  containing  both  odd  and  even  substitutions.  There  are  likewise  two 
groups  isomorphic  to  (J.1J.2 -43^.4^.5)20,  one  generated  by  u,  the  other  by  a]al.u. 
The  first  of  these  groups  contains  odd  substitutions,  the  second  only  even,  (xggggo, 
is  contained  self-conjugately  in  both. 

Only  one  group  G'  can  be  found  for  this  head,  as  no  new  v^  or  vl  fulfills  the 
necessary  conditions.  Such  a  substitution  would  necessarily  be  of  the  form  gv 
or  Gv',  where  g  would  belong  to  the  group  -07776-  If  cr  were  even,  the  group  so 
generated  would  be  a  repetition  of  the  group  generated  by  v  and  v'.  If  cr  were 
odd,  the  cubes  of  at-,  gv'  would  not  be  contained  in  the  head. 

Two  groups  can  be  found  isomorphic  to  (Ai^A^A^A^A^)  all,  the  substitutions 
ic  and  lu'  generating  one  group,  the  substitutions  alal.iv ,  o}ai.  ic'  generating  the 
other.     This  latter  group  contains  only  even  substitutions. 

From  this  head  we  have,  therefore,  derived  eight  groups  : 

j^gggg,  s\  of  order  19440, 

{^jggg,  s.  t\  of  order  38880., 

^Sgggg,  s,  a\al.t\  of  order  3888O3, 

{Hgggg.s,  u\  of  order  7666O2, 

^^gggg.s,  a\al.u\  of  order  7666O3, 

■J-S3888.  ^.  "y't  of  order  233280, 

jSgggg,  w,  iv'\  of  order  4665602, 

■IjBgggs,  a\al.w,  a\al.w'\  of  order  46656O3. 

III.  5486  furnishes  us  with  only  one  group  isomorphic  to  (^1^.2  ^3 ^4^5)  eye, 
for  an  examination  of  the  groups  given  by  all  possible  types  of  substitutions  s^ 
shows  that  each  of  these  groups  is  merely  the  group  generated  by  the  help  of 
s  and  transformed  with  respect  to  some  easily  discovered  substitution.  More- 
over, there  is  but  one  group  isomorphic  to  (J.i  ^12^3^4^5)10'  viz.,  that  generated 
with  the  help  of  t.  Any  cofactor  of  t  must  be  of  one  of  the  types  a\a}2,  a\al .  a^al , 
a\al.alal.a\al,  ajal .  alaj .  a{al .  a\al ,  but  any  t^  got  by  means  of  these,  transforms 


Fifteen  and  the  Primitive  Substitution   Groups  of  Degree  Eighteen.       13 

s  into  5^  (a  substitution  not  in  the  head).  Precisely  the  same  reasoning  shows 
that  there  is  only  the  one  group  isomorphic  to  (^A-^A.;^A^A^A^).,(^. 

In  addition  to  the  group  isomorphic  to  (^1^2^.3  ^4^.5)  pos  generated  by 
means  of  the  substitutions  v  and  v',  we  must  examine  groups  generated  with  the 
help  oi  v^  and  v'^,  substitutions  which  contain  as  cofactors  of  v,  v'  respectively 
the  products  of  transposition,  one  transposition  from  each  system.     A  number 
of  these  may  be  rejected  at  once,  but  we  are  left  with  the  possible  forms  : 
■yj  =  a\a\ .  a\a\ .  v  =  alala^  .  alalcil .  alalal , 
V2  ==  ala] .  a^ai .  alal .  v  =  <i\alal .  alalal .  a\a\ .  afa| , 
v[  ==  a\a\ .  a\a\ .  v'  =  alajal .  cdalal .  alalal , 
vi  =  a\al .  a^dl .  afal .  v'  =  a\aial .  alalal .  a\dl  .a\a\. 

But,  since  vi,  -yg  ^-i'^  transformable  into  v,  and  v[,  v'^  are  transformable  into  v',  it  is 
impossible  to  generate  any  group  by  means  of  any  combination  of  these  four 
substitutions  excepting  a  group  that  can  be  transformed  into  the  one  generated 
by  means  of  v  and  v' . 

A  similar  examination  of  all  groups  isomorphic  to  (^1^0^-3-44^5)  all,  shows 
that,  in  addition  to  the  group  generated  with  the  help  of  id  and  w\  there  is  one 
other  group  generated  by  means  of  w^  =  alal .  lo  and  iv'. 

From  this  head  are  therefore  formed  the  six  following  groups : 

]  1/486,  «1  of  order  2430i , 
{ffise,  s,t\  of  order  4860i, 
{^^486)  s ,  u\  of  order  9720, 
)/7,8c,v,  v'\  of  order  29160i, 
jZ/jse,  w,  iv'\  of  order  58320i, 
)  £?436 ,  alal  .w,io'\  of  order  58320o. 

IV.  7/243  gives  one  group  isomorphic  to  {A^AoA^A^A^)  eye,  by  means  of  s. 
The  only  other  permissible  forms  of  5„  are  of  the  type 

§1  =  ala] .  alal  .s,     53  =  a}aj .  a^al .  alal .  a\al .  s . 

But  s^  and  s^  are  each  the  transformed  of  s  with  respect  to  some  substitution 
that  transforms  the  head  into  itself;  therefore,  there  is  only  the  one  group  of  this 
type.  On  the  other  hand,  there  are  two  groups  isomorphic  to  (J-j  Jo ^3^4^5)10, 
since  both  t  and  t^  =  a\al .  ajal .  alal .  alal .  alal .  t  fulfill  all  necessary  conditions  and 
generate,  one  a  group  of  even  substitutions,  the  other  a  group  containing  odd 


14  Martin  :    On  the  Imprimitive  Substitution   Groups  of  Degree 

substitutions.  There  are  also  two  groups  isomorphic  to  {A-^A^A^A^A^oq,  one 
containing  both  odd  and  even  substitutions,  the  other  only  even.  These  are 
generated  respectively  by  means  of  u  and  of  11^:=  a\a\.aial.a\al.a\cd.a\a.\.u, 
and  each  contains  as  a  self-conjugate  subgroup  the  group  isomorphic  to 
(^1^.0^3^4^5)10  that  consists  entirely  of  even  substitutions. 

Only  one  group  can  be  found  isomorphic  to  {A-^A^A^A^A-^  pos,  and  this  is 
the  one  formed  by  the  help  of  v  and  v'.  An  examination  of  the  various  substi- 
tutions v^  and  v'^  corresponding  to  various  types  of  cofactor  of  i?  and  v'  shows 
that  all  groups  formed  by  means  of  these  substitutions  are  transformable  into 
the  one  group. 

On  the  other  hand,  we  have  two  distinct  groups  corresponding  to 
{A^AzA^A^A^  all,  the  one  consisting  of  both  odd  and  even  substitutions  and 
generated  by  the  aid  of  ic  and  u:',  the  other  consisting  entirely  of  even  substitu- 
tions and  generated  by  the  aid  of  a\al .  ic  and  a\al .  iv'. 

From  this  head  we  have,  therefore,  the  eight  following  groups : 

{Has,  s\  of  order  1215, 

1^243,  s,  t\  of  order  2430o, 

]  ^043 ,  s,  a\al . rqal . alal .  a\cd  .  a\c4  .t\  of  order  243O3 , 

Jj^243>  *>  ^^(  of  order  486O2, 

]  5o43,  s,  alal .  a\al .  alal .  alai .  a^al  .u\  of  order  486O3, 

\R2i3,v,  v'\  of  order  14580, 

\Eziz,  w,  w'\  of  order  291600, 

"I  ^043 ,  alal .  ic,  alal .  w'  j  of  order  29 1 6O3 . 

V.  R^  furnishes  one  group  corresponding  to  each  transitive  group  of  degree 
5.  These  groups  are  generated  respectively  by  the  substitutions  s,  t,  ic,  v,  v',  ic,  to', 
and  can  readily  be  seen  to  be  indentical  with  those  of  orders  30o,  6O2,  120,  360o, 
720  included  among  the  groups  with  three  systems  of  imprimitivity.  An  inter- 
change of  suffixes  and  indices  in  the  one  set  of  groups  gives  the  generating  sub- 
stitution of  the  other  set  of  groups. 

VI.  j^g  furnishes  groups  corresponding  to  the  transitive  groups  of  degree  5 
by  means  of  the  substitutions  s,  t,  u,  v,  v',  ?r,  iv'.  As  in  the  last  case,  however, 
these  correspond  to  the  groups  of  orders  15,  30i,  6O1,  180,  360i  included  in  the 
groups  with  three  systems  of  imprimitivity.     By  the  use  of  the  cofactor  a  =  a]al . 


Fifteen  and  the  Primitive  Suhstitution   Groups  of  Degree  Eiglitcen.       15 

alal .  a\al .  a\al .  a\al  three  more  groups  can  be  found  generated  respectively  by  the 
help  of  t]^  =  at,  ill  =  <7^*.  ^^1  =  cr^^)  ^  =  cr^^'-  These  groups,  however,  are  seen 
to  be  identical  with  those  of  orders  SOg,  60g,  and  SGOg  included  in  the  groups 
with  three  systems  of  imprimitivity.  This  head  gives,  therefore,  no  group  essen- 
tially new. 

VII.  In  the  discussion  of  the  head  unity  a  useful  theorem  is  the  following 
given  by  Frobenius  (Crelle  t.  ci,  p.  287): 

The  average  number  of  elements  in  all  the  substitutions  of  a  group  is  n  —  a,n 
being  the  degree  of  the  group,  and  a  the  number  of  its  transitive  constituents. 

The  only  transitive  groups  of  degree  5  containing  15  as  a  factor  of  the  order 
are  the  symmetric  and  alternating  groups.  We  have  therefore  to  find  an  imprira- 
itive  group  of  degree  15  with  5  systems  of  intransitivity  simply  isomorphic  to 
the  alternating  (symmetric)  group  in  5  letters. 

In  determining  the  imprimitive  group  corresponding  to  {A-^A^A^A^A^)  pos, 
we  make  use  of  the  following  facts:  (1)  the  15  conjugate  substitutions  corres- 
ponding to  terms  of  the  type  A^A^.  J.3j44must  be  of  degrees  12  or  14;  (2)  the  20 
conjugate  substitutions  corresponding  to  terms  of  the  type  A^  A^  A^  must  be  of 
degrees  9,  12,  or  15;  (3)  the  24  conjugate  substitutions  corresponding  to  terms 
of  the  type  A^A^A^A^A^  must  be  of  degree  15.  It  must,  therefore,  be  possible  to 
solve  the  equation 

15  (12  +  2a)  +  20  (9  +  S(3)  -f  24.15  =  14.60 

where  a  =0,  1 ;  /3  =  0,  1,  2.     The  only  solution  is  a  =  0,  /3  =  2. 

Therefore  the  imprimitive  group  we  are  seeking  contains  among  its  substi- 
tutions 15  of  degree  12  and  order  2,  20  of  degree  15  and  order  3,  24  of  degree  15 
and  order  5.  Making  use  of  the  relations  among  the  generating  substitutions  of 
such  a  group  of  order  60  as  given  in  Burnside,  Theory  of  Groups,  p.  107,  we  find 
that  the  two  substitutions  corresponding  to  ^.1-43^3^.^^.5,  A^^A-yA^A},  substitutions 
which  will  generate  (^1-43^.3^4^5)  pos,  are  respectively , 

s  =z  a\alalalal .  alalalalal .  alalala^al, 

12  3     4  12  34  12  34 

p  =  a\a^ .  alal .  at^ai .  a^a} .  a^a^ .  ala] ; 

s  and  p  are  therefore  the  generating  substitutions  of  an  imprimitive  group  simply 
isomorphic  to  the  alternating  group  of  degree  5. 

In  determining  a  group  simply   isomorphic  to  {AiAoA^A^A^)  all,  we  argue  as 
before  in  regard  to  the  various  sets  of  conjugate  substitutions.     The  15  substitu- 


16  Martin  :    On  the  Imprimitive  Substitution   Groups  of  Degree 

tions  corresponding  to  terms  of  the  type  A^A^ .  A^A^  are  of  degrees  12  or  14,  the 
20  corresponding  to  the  type  A^A^^A^,  are  of  degrees  9,  12,  or  15,  the  24  corres- 
ponding to  the  type  A^A^A^A^A^  are  of  degree  15,  the  10  corresponding  to  the 
type  A-^A^  are  of  degrees  6,  8,  10,  or  12;  the  30  corresponding  to  the  type  A-^A^ 
^3^4  are  of  degrees  12  or  14;  the  20  corresponding  to  the  tj^pe  A-^AoA^A^A^^yq  of 
degree  15.     The  equation  to  be  satisfied  is  therefore 

15  (12  +  2a)  +  20  (9  -f  3,5)  +  24.15  +  10  (6  +  2/)  4-  30  (12  +  2.^)  -f  20.15 

=  14.120  where  a  =  0,  l;/3  =  0,  1,  2;  ^  =  0,  1,  2,  3;  ^  =  0,  1. 

The  only  solution  is  a  =  0,  /3  =  2,  ^  =  3,  (5  =  1.  The  substitutions  A^A^A^A^A^, 
A^A^A^Ar^,  A^A^A^A^  will  generate  the  group  {A^A^A^A^A-^  all,  and  corresponding 
to  these  as  generators  of  the  imprimitive  group  we  have  the  three   substitutions, 

i^SiT        1-^34';        i-'t4'; 

5  rr  a{aiala\al .  aMp,Uqal .  a^aia'^alcq , 
G  =  alal .  alalalaj .  ao«|a|rt| .  alalalai, 
p  =r  a\al .  alal .  alal .  alal .  ajal .  ajaf . 

To  sum  up  the  results  of  the  preceding  work,  the  16  heads  with  three  sys- 
tems of  intransitivity  give  41  groups  with  three  systems  of  imprimitivity.  The 
7  heads  with  five  systems  of  intransitivity  give  42  groups  with  five  systems  of 
imprimitivity,  but  of  these  13  groups  contain  also  three  systems  of  imprimitivity. 
Therefore  there  are  70  imprimitive  groups  of  degree  15  as  determined  in  this 
paper. 

Primitive  Substitution  Groups  of  Degree  Eighteen. 

The  main  theorems  employed  in  this  investigation  of  primitive  groups  are 
the  following,  in  which  p  is  always  to  stand  for  a  prime  number. 

I.      The  order  of  a  primitive  group  of  degree  n  cannot  exceed ' ,  where 

2,  3,  .  .  .  .  p  are  the  distinct  primes  ichich  are  less  than  f  n.  (Burnside,  Theory  of 
Groups,  p.  199). 

n.  A  group  of  degree  p  -\-  x  or  of  degree  2p  -\-  x,  x  >»  2,  cannot  be  more  than 
X  times  transitive.     (Miller,  Bull.  A.  M.  S.,  v.  IV,  pp.  142,  143). 

III.     If  a  primitive  group  of  degree  n  contains  a  circidar  substitution  of  prime 


Fifteen  and  the  Primitive  Substitution   Groups  of  Degree  Eighteen.       17 

order  p,  the  group  is  at  least  {n — p -\- \)-fold  transitive.     (Cole's  tr.  of  Netto's 
Theory  of  Substitutions,  p.  93). 

IV.  A  self-conjugate  subgroup  of  a  primitive  group  must  he  transitive.  (Burn- 
side,  1.  c,  p.  187). 

V.  A  self -conjugate  stibgroitp  of  a  x-ply  transitive  group  of  degree  n{2<C  x  <Cn) 
is  hi  general  at  least  (x  —  ^)-p^y  ti'arisltive.  The  only  excepticm  is  that  a  triply  tran- 
sitive group  of  degree  2"'  may  have  a  self-conjugate  subgroup  of  order  2™.  (Burn- 
side,  1.  c,  p.  189). 

VI.  A  group  G  ivhich  is  at  least  doubly  travisitive  either  must  be  simple  or  mus 
contain  a  simple  group  H  as  a  self-conjugate  subgroup.  In  the  latter  case  no  opera 
tion  of  G  except  identity  is  per  mutable  ivith  every  operation  of  II.  The  only  exceptions 
to  this  statement  are  that  a  triply  transitive  group)  of  degree  2'"  may  have  a  self 
conjugate  subgroup)  of  order  2"\  and  that  a  doubly  transitive  group  of  deyree  p^  may 
have  a  self -conjugate  subgroup  of  order  p'".     (Burnside,  1.  c,,  p.  192). 

VII.  The  substitutions  of  a  transitive  group  G  which  leave  a  given  symbol 
unchanged  form  a  maximal  subgroup  G^ ,  ivhich  is  one  of  a  set  of  7i  conjugate  sidy- 
groups,  each  leaving  one  of  the  n  elements  unaffected.     (Burnside,  1.  c,  p.  140). 

VIII.  The  number  of  substitutions  of  degree  l<C,n  contained  in  a  transitive 
group)  of  degree  n  is  equal  to  the  number  of  substitutions  of  this  same  degree  I  contained 

in  the  maximal  subgroup  G^  of  degree  n  —  1  multipliedby  ■ j- .     (Stated  by  Miller, 

Quar.  Jour.  of.  Math.  v.  XXVIII,  p.  215.) 

IX.  The  average  number  of  elements  in  all  the  substitutions  of  a  group  is  n  —  a, 
n  being  the  degree  of  the  group  and  a  the  number  of  its  transitive  constituents.  (Fro- 
benius,  Crelle,  t.  ci,  p.  287.) 

X.  Sylow's  theorem,  as  stated  by  Burnside,  1.  c,  p.  92,  or  by  Sylow,  "Theo- 
remes  sur  les  groupes  de  substitutions,"  Math.  Ann.,  v.  V  (1872),  pp.  584  et  seq. 

XI.  The  class  of  a  primitive  group  of  degree  n  is  the  same  as  the  class  of  its 
maximal  subgroup  that  leaves  one  element  unaffected. 


18  Martin  :    On  the  Imiyrimitive  Substitution  Groups  of  Derjree 

While  the  preceding  theorems  are  used  throughout  the  work  on  primitive 
groups,  the  following  are  used  mainly  in  the  determination  of  simply  transitive 
primitive  groups. 

XII.  A  simply  transitive  primitive  group)  G  of  degree  n  cannot  contain  a  transi- 
tive subgroup  of  degree  Jess  than  n.  (Miller,  Quar.  Jour,  of  Math.,  v.  XXYIII, 
p.  215. 

XIII.  When  Gi  contains  a  self -conjugate  suhgj'oup  H  of  degree  n  —  a,  R  must 
be  intransitive,  and  it  must  be  the  transform  with  respect  to  substitutions  of  G  of  any 
one  of  a —  1  other  subgroups  of  G^  (S[,  HL,  ....  Hl_-^ .  (Miller,  Proc.  Lon. 
Math.  Soc,  v.  XXVIII,  p.  534.) 

XIV.  All  the  prime  numbers  which  divide  the  order  of  one  of  the  transitive  con- 
stituents of  Gi  divide  also  the  orders  of  each  of  the  other  transitive  constituents. 

Corollary  I.  If  one  of  the  transitive  coivstituents  of  Gi  is  of  a  prime  degree, 
each  of  its  other  transitive  constituents  is  of  the  same  or  a  larger  degree,  and  the 
order  of  G^  is  the  same  as  the  order  of  the  group  formed  by  these  other  transitive 
constituents. 

Corollary  II.  If  the  order  of  Gi  is  not  divisible  by  the  square  of  a  prime 
number,  all  its  transitive  constituents  are  of  the  same  order,  and  G^  is  formed  by 
establishing  a  simple  isomorphism  between  them.     (Miller,  1.  c,  p.  536.) 

XV.  If  a  transitive  constituent  of  Gx  is  of  a  prime  order,  the  order  of  G^  is  the 
same  prime  number,  and  G  is  of  class  n  —  1 . 

Corollary.  If  Gi  contains  a  constituent  of  degree  2,  its  order  is  2,  and  the  degree 
of  G  is  a  prime  number.     (Miller,  1.  c,  p.  536.) 

The  above  theorems  are  given  in  the  form  and  with  the  symbols  most  con- 
venient for  use,  and  so  are  not  always  exact  quotations  from  the  papers  and 
books  referred  to,  while  the  references  given  are  not  always  references  to  the 
original  paper  in  which  the  theorem  appeared. 

Applying  these  theorems  now  to  the  special  case  in  ^vhich  7i  =  18  ,  we  pro- 
ceed as  follows  : 

Since  18  =  2.  7  +  4,  by  Theorem  II  a  primitive  group  cannot  be  more  than 
4-ply  transitive. 


Fifteen  and  the  Primitive  Substitution   Groups  of  Degree  Eighteen.       19 

By  Theorem  I  the  order  is  seen  not  to  exceed 

18! 


2.3.5.7.11 


=  2^13^517.13.17 


If  the  group  included  circular  substitutions  of  orders  2,  3,  5,  7,  11,  13,  it  would 
be  at  least  17,  16, 14, 12,  8,  6-fold  transitive  respectively  according  to  Theorem 
III.  This  is  impossible;  therefore  circular  substitutions  of  these  orders  are  not 
present,  and  consequently  we  see  at  once  that  11  and  13  cannot  be  factors  of 
the  order. 

If  the  order  includes  the  factor  7,  then,  by  Theorem  X,  there  is  a  subgroup 
of  order  7.  This  must  consist  of  the  powers  of  a  substitution  composed  of  two 
cycles  of  7  elements  each,  and  it  must  be  contained  self-conjugately  in  a  group 
of  order  7  .  4m  that  interchanges  transitively  among  themselves  the  four  remain- 
ing elements.  (Cf.  Burnside,  Theory  of  Groups,  p.  202.)  It  is  quite  possible 
to  establish  a  (7a,  1)  isomorphism  between  an  imprimitive  group  of  degree  14 
with  the  systems  of  imprimitivity  7,  7  and  a  transitive  group  of  degree  4;  there- 
fore 7  may  be  a  factor  of  the  order. 

A  subgroup  of  order  5^  cannot  be  present,  as  it  would  have  to  be  intransi- 
tive with  the  systems  of  intransitivity  5,  5  or  5,  5,  5.  In  the  one  case,  it  would 
have  to  be  contained  self-conjugately  in  a  group  of  order  5l  8m,  in  the  other,  in 
a  group  of  order  5^.  3 .  772 .  In  either  case,  a  circular  substitution  of  order  5  would 
be  present,  which  is  impossible. 

The  factor  5  may  be  contained  in  the  order,  as  it  is  possible  to  establish  a 
(5,  1)  isomorphism  between  the  cyclical  group  of  degree  15  and  the  cyclical 
group  in  the  remaining  three  letters. 

The  order  must,  therefore,  be  a  factor  of  2^^.  3^  5  .  7  .  1 7. 

Simply   Transitive  Groups. 

The  maximal  subgroup  G^  that  leaves  a^  unaffected  is  intransitive  (Theorem 
XII),  and  its  order  is,  therefore,  a  factor  of  2^^  3l  5.7.  Moreover,  its  class  can- 
not be  less  than  6,  for  if  it  were  2,  3  or  5,  G^  would  necessarily  contain  a  transi- 
tive subgroup  of  too  low  a  degree,  and  it  cannot  be  of  class  4,  if  G  is  to  be 
primitive.     (Netto,  1.  c,  p.  138.) 

By  Theorem  XIV,  Cor.  I,  it  is  evident  that  Gi  cannot  contain  a  transitive 
constituent  of  degree  13  or  11;  by  Theorem  XIV   it  cannot  contain  a  transitive 


20  Martin  :   On  the  Imprimitive  Substitution   Groups  of  Degree 

constituent  of  degree  15  or  14,  and  by  Theorem  XV,  Cor.,  it  cannot  contain  a 
constituent  of  degree  2 . 

If  one  of  the  transitive  constituents  of  G^  is  of  degree  1 2,  the  other  must  be 
of  degree  5.  The  isomorphism  between  the  transitive  groups  of  degrees  12  and 
5  must  be  an  (a,  1)  isomorphism,  where  a  itself  may  be  equal  to  1.  By  Theorem 
XIV,  the  order  of  the  group  of  degree  5  must  contain  the  factors  5,  3,  2;  there- 
fore this  group  must  be  either  the  alternating  or  the  symmetric  group  of 
degree  5.  If  the  isomorphism  is  more  than  simple,  then  the  group  of  degree  12 
must  be  an  imprimitive  group  with  6  systems  of  imprimitivity.  The  head  for 
such  an  imprimitive  group  as  we  require  is  the  intransitive  group  of  order  2 
and  degree  12  given  by  [l,  aiotg  •  ^3^*4  •  %<^6  •  ^t'^s  •  o^9<^io  •  ^u^d  •  The  group 
(rti3ai4«i5«i6'^i7)  pos  contains 

24  conjugate  substitutions  of  order  5  and  degree  5, 
20  conjugate  substitutions  of  order  3  and  degree  3, 
15  conjugate  substitutions  of  order  2  and  degree  4. 

Corresponding  to  these  in  the  group  of  degree  17,  we  have  1  substitution  of 
degree  12  and  order  2,  24  of  degree  15  -f-  2a  and  order  5  or  10,  24  others  of 
degree  15+  2a'  and  order  5  or  10,  20  of  degree  15  +  2(5  and  order  3  or  6, 
ogether  with  20  of  degree  15+  2/3'  and  order  3  or  6,  15  of  degree  12  +  ly 
and  order  2  or  4,  and  15  of  degree  12  +  2^'  and  order  2  or  4,  where 

a  =  0,  1  ;  a' =  0,  1  ;  /^  =  0,  1  ;  ;^'  =  0,  1  ;  7  =  0,  1,  2;  /  =  0,  1,  2. 

By  Theorem  IX,  the  following  equation  must  be  satisfied  : 

12  +  24(15  +  2a)  +  24(15  +  2a')  +  20(15  +  2/3)+  20(15  +  2/?') 

+  15  (12  +  2y)  +  15  (12  +  2/)  =  120  .  15. 

The  only  type  of  solution  is  given  by  a  =  /?  =  ^  =  |5'  =  0 ,  a'  =  1 ,  }^'  =  2  .  6^^, 
therefore,  contains  both  a  self-conjugate  subgroup  of  degree  12  and  order  2,  and 
15  conjugate  subgroups  of  the  same  type.  But,  by  Theorem  XIII,  only  5  such 
conjugate  subgroups  should  exist  if  this  group  is  to  be  the  G^  of  a  simply  transi- 
tive primitive  group.  This  intransitive  group  gives  us,  therefore,  no  such  group 
as  we  require. 

For  precisely  the  same  reason  the  intransitive  group  formed  by  establishing 
a  (2,  1)  isomorphism  between  an  imprimitive  group  of  degree  12   and  order  240, 


Fifteen  and  the  Primitive  Sahstitution  Groups  of  Degree  Eighteen.       21 

and  the  symmetric  group  of  degree  5  cannot  be  employed  in  the  formation  of  a 
simply  transitive  primitive  group  of  degree  18. 

If  the  isomorphism  is  simple,  the  group  G^,  including  the  alternating  group 
of  degree  5,  must  contain  24  substitutions  of  degree  10  or  15  and  order  5,  20  of 
degree  6,  9,  12  or  15  and  order  3,  15  of  degree  6,  8,  10, 12,  14  or  16  and  order  2. 
The  following  equation  must,  therefore,  be  satisfied  :  24  (10  +  a)  +  20  (6  + /^) 
+  15  (6  +  ^)  =  60  X  15,  where  a  =  0,  5 ;  ^  =  0,  3,  6,  9  ;  ^  =  0,  2,  4,  6,  8,  10. 
The   only  solution  is  a  =  5,  /i?  =  9,  )/  =  10.      6^^  is,  therefore,  of  class  15. 

A  group  G  of  degree  18,  formed  with  the  help  of  this  6^^  and,  therefore,  of 
order  60.  18,  contains  36  conjugate  subgroups  of  order  5,  each  of  which  is  con- 
tained self-conjugately  in  a  group  of  order  30.  As  each  of  these  subgroups  of 
order  5  is  already  self-conjugate  in  a  group  of  order  10,  the  construction  of  the 
generating  substitutions  of  such  a  group  is  an  easy  matter.      G^  is  generated  by 

and  by  f  =  a^a^ .  a^a^ .  a^cti^ .  o^a^ .  a^ai^ .  a^a-^^ .  ai3«i5 .  aua^^ , 

and  contains,  as  one  of  the  above-mentioned  groups  of  order  10,  the  group  gene- 
rated by 

V  =  s~Us  =  a^a^ .  a^a^ .  a^a^ .  a^a^ .  agajg .  aio«ii .  ctisdi^  •  «i4«i6. 

The  group  \u,  v\  is  a  subgroup  of  a  group  of  degree  18  and  order  30  formed 
by  establishing  a  (5,  1)  isomorphism  between  an  imprimitive  group  of  degree  15 
and  order  30  with  u  and  its  powers  as  head  and  the  symmetric  group  in  the  three 
elements  a^a^a^^.  The  question  then  reduces  to  that  of  the  determination  of  a 
substitution  of  degree  18  and  order  3  that  will  transform  the  head  \u\  into  itself, 
interchange  cyclically  the  three  systems  of  \u\,  and  be  in  its  turn  transformed 
into  its  square  by  v.  An  examination  of  all  substitutions  fulfilling  these  condi- 
tions results  in  finding  none  that  do  not  give,  when  combined  with  other  substi- 
tutions of  Gi,  substitutions  that  cannot  possibly  belong  to  a  simply  transitive 
primitive  group  containing  Gi  as  a  maximal  subgroup. 

There  is  no  primitive  or  imprimitive  group  of  degree  12  simply  isomorphic 
to  the  group  [ahcdef) ^20',  consequently,  no  isomorphism  can  be  established 
between  the  symmetric  group  of  degree  5  and  a  transitive  group  of  degree  12. 

There  remains  the  question  whether  the  symmetric  group  of  degree  5  can 
be  put  in  a  simply  isomorphic  relation  to  one  of  the  imprimitive  groups  of  degree 


22  Martin  :   Oti  the  Imprimitive  Substitution   Groups  of  Degree 

12  and  order  120  that  have  both  six  and  two  systems  of  imprivitivity.  Such 
o^roups  of  degree  12,  however,  contain  two  self-conjugate  subgroups  of  orders  2 
and  60  respectively,  and,  therefore,  are  not  in  a  simply  isomorphic  relation  to 
the  symmetric  group  of  degree  5. 

If  one  of  the  transitive  constituents  is  of  degree  10,  the  other  can  only  be  of 
degree  7.  By  Theorem  XIV,  the  group  of  degree  10  must  contain  7  as  a  factor 
of  its  order,  and,  therefore,  must  be  either  the  alternating  or  the  symmetric 
group.  It  is  impossible  to  establish  an  isomorphic  relation  between  either  of 
these  groups  and  one  of  degree  7  without  introducing  substitutions  of  too  low  a 
degree. 

If  one  of  the  transitive  constituents  is  of  degree  9,  the  remaining  constituent 
may  be  either  intransitive  in  two  systems  of  four  elements  each  or  intransitive  in 
eight  elements.  The  isomorphism  can  in  neither  case  be  simple,  as  an  examina- 
tion of  all  groups  of  degree  8  and  orders  equal  to  those  of  transitive  groups  of 
degree  9  shows  that  in  each  case  a  system  of  intransitivity  of  degree  2  enters, 
with  the  single  exception  of  a  group  of  order  144.  Here,  however,  the  group  of 
degree  9  contains  a  substitution  of  order  8,  while  an  inspection  of  the  corres- 
ponding groups  of  degree  8  shows  no  substitution  of  that  order. 

The  isomorphism  is,  therefore,  an  (a,  p)  isomorphism,  where  a  and  /3  are 
not  simultaneously  equal  to  one. 

When  neither  a  nor  p  is  equal  to  one,  G^  must  be  formed  from  an  imprimi- 
tive  group  of  degree  9  and  an  intransitive  group  of  degree  8.  The  order  of  each 
transitive  constituent  must  contain  3  as  a  factor,  and,  therefore,  the  group  ot 
degree  8  must  be  some  combination  of  the  alternating  and  symmetric  groups  of 
degree  4  in  two  systems  of  elements.  The  only  combinations  possible,  consistent 
with  the  requirements  of  class,  are  got  by  establishing  a  simple  isomorphism 
between  the  two  symmetric  groups  of  degree  4  or  between  the  two  alternating 
groups  of  the  same  degree.  Every  relation  of  isomorphism  established  between 
these  groups  of  degree  8  and  any  impriraitive  groups  of  degree  9  consistent  with 
the  requirements  of  class,  results  in  a  Gi  that  contains  a  self-conjugate  subgroup 
of  order  4  and  degree  8,  and  no  other  subgroups  of  the  same  order.  This  case, 
therefore,  gives  no  simply  transitive  primitive  group. 

When  a  becomes  1,  the  group  of  degree  8  must,  as  before,  be  composed  of 
either  the  symmetric  or  the  alternating  groups  of  degree  4  in  two  sets  of 
elements  put  into  the  relation  of  simple  isomorphism.  The  order  of  such  a  group 
does  not  contain  9  as  a  factor;  therefore   this  case  gives  no  possible  G^. 


Fifteen  and  the  Primitive  Substitution  Groups  of  Degree  Eighteen.        23 

When  /?  becomes  1,  the  group  of  degree  9  must  be  impriraitive.  No  tran- 
sitive group  of  degree  8  stands,  however,  in  the  given  relation  of  isomorphism 
towards  an  imprimitive  group  of  degree  9.  The  only  permissible  intransitive 
groups  of  degree  8  are  combinations  of  the  symmetric  and  alternating  groups  of 
degree  4  in  two  sets  of  elements,  and  none  of  these  are  isomorphic  in  the  given  way 
to  any  imprimitive  group  of  degree  9. 

If  one  of  the  transitive  constituents  is  of  degree  8,  we  may  have  the  systems  8, 
6,  3  or  8,  3,  3,  3.  In  both  cases  we  have  an  (a,  1)  isomorphism  between  an 
intransitive  group  of  degree  14  and  the  symmetric  group  of  degree  3.  The  group 
of  degree  8  is  not  primitive,  as  no  suitable  isomorphic  relation  can  be  established 
between  a  primitive  group  of  degree  8  and  an  imprimitive  or  an  intransitive  group 
of  degree  6.  The  only  imprimitive  groups  of  degree  8  that  can  be  used  are  those 
with  the  head  (1,  aia^-dia^ .  OrfitQ  •a.^cig)  that  are  isomorphic  to  a  group  of  degree 
4  and  order  1 2  or  24.  Such  groups,  however,  cannot  be  combined  with  the  groups 
in  the  remaining  9  elements  in  such  a  way  as  to  generate  a  group  capable  of  being 
the  Gi  of  one  of  the  required  primitive  groups. 

The  case  in  which  Gi  contains  a  transitive  constituentof  degree  7  has  already 
been  discussed,  as  according  to  Theorem  XIV,  Cor.  I,  the  remaining  constituents 
must  be  of  larger  degree. 

If  Gi  contains  a  transitive  constituent  of  degree  6,  the  systems  may  be  either 
6,  6,  5  or  6,  4,  4,  3.  For  the  former  system  the  only  possible  arrangement  is  to 
establish  a  simple  isomorphism  between  the  three  groups  {aia./t^a^ar^aQJQQ ,  {a^agaya^Q 
«n«i3)6o»  (^I3f«i4%5«^i6^j7)  pos,  or  between  the  groups  {axa.^a.ja^a^aQ\c,Q,  («7«8«/'io«ii«i2)  leo- 
(ajgai^aisaieai,)  all.  An  examination  of  the  two  groups  (rj  formed  from  these  iso- 
morphisms shows  that  these  are  not  the  maximal  subgroups  of  simply  transitive 
primitive  groups  of  degree  18. 

If  the  systems  are  6,  4,  4,  3,  only  the  imprimitive  groups  of  degree  6,  the 
alternating  and  symmetric  groups  of  degree  4,  and  the  symmetric  groups  of  degree 
3  are  involved.  The  group  formed  by  the  system  6,  4,  4  has  an  (a,  1)  isomorphism 
to  the  group  of  degree  3,  and  this  isomorphism  cannot  be  simple.  No  combination 
of  these  groups  can  be  found  fulfilling  all  the  necessary  conditions. 

The  case  in  which  Gi  contains  a  transitive  system  of  degree  5  has  already 
been  discussed,  as  the  remaining  systems  must  be  of  degree  greater  than  5. 

If  (?!  contains  a  transitive  system  of  degree  4,  the  only  arrangement  possible 
is  4,  4,  3,  3,  3.  The  groups  involved  are  therefore  the  symmetric  groups  of  degree 
3  and  4,  and  the  alternating  group  of  degree  4.     One  group  consistent  with  the 


24  Martin  :    On  the  Imprimitive  Suhstitution   Groups  of  Degree 

requirements  of  class   is  got  by  establishing  a  (1,4)   isomorphism  between  the 
group, 

This  group  contains,  however,  one  and  only  one  subgroup  of  degree  8  and  order  2. 

A  second  group  is  got  by  first  establishing  a  (4,  1)  isomorphism  between  {a^a^a 
a^.a^a^a^a^  all  and  (orgajoaji)  all;  and  then  establishing  a  (12,  3)  isomorphism  between 
the  group  of  order  24  so  formed  and  the  group  {a^^a^^a^^ .  ais^ie^n)  all.  This  group 
of  degree  17  contains  only  one  subgroup  of  order  4  and  degree  8;  therefore  it 
cannot  become  a  G^ . 

Gi  cannot  contain  onl}^  systems  of  degree  less  than  four,  as  in  such  a  case 
a  system  of  degree  2  would  have  to  enter. 

There  is,  therefore,  no  simply  transitive  primitive  group  of  degree  18.  This 
result  when  joined  to  all  other  determinations  of  similar  groups  shows  that  there 
is  no  simply  transitive  primitive  group  of  degree  p  +1,  i>  a  prime  number  and 
<17. 

Multiply  transitive  groups. 

Among  the  transitive  groups  of  degree  17  five  contain  a  self-conjugate  sub- 
group of  order  17.  These  are  of  order  17,  2.17,  4.17,  8.17,  16.17  respectively, 
while  all  excepting  the  first  are  of  class  16. 

If  a  primitive  group  of  degree  18  and  order  18.17  existed,  such  agroup  would 
contain  18  conjugate  subgroups  of  degree  17.  It  would  therefore  contain  17  sub- 
stitutions of  degree  18  and  18.16  of  degree  17.  By  Sylow's  theorem  since  18.17 
=:  2.3-.  17,  such  a  group  contains  either  1  or  34  subgroups  of  order  3~.  A  sub- 
group of  this  order  must  be  intransitive,  therefore  cannot  be  self  conjugate,  and 
it  is  impossible  to  form  34  subgroups  of  order  9  from  17  substitutions  of  degree  18. 
No  such  group  of  degree  18  exists. 

A  primitive  group  of  degree  18  and  order  18. 17.2  =  2l3^  17  would  contain 
among  its  substitutions  153  of  class  16  and  order  2,  288  of  class  17and  order  17, 
170  of  class  18.  This  group  must  contain  either  1,  4,  or  34  conjugate  subgroups 
of  order  3'^  As  before,  a  subgroup  of  this  order  cannot  be  self-conjugate,  as 
it  is  intransitive.  If  there  were  4  conjugate  subgroups,  each  would  be  self- 
conjugate  in  a  group  of  order  3l  17  involving  all  18  letters  and  necessarily  tran- 
sitive. Such  a  group  is  non-existent.  If  there  were  34  conjugate  subgroups 
they  must  be  of  degree  18,  and  there  are  not  enough  substitutions  of  class  18  to 
form  all  these  subgroups. 


Fifteen  and  the  Primitive  Substitution  Groups  of  Degree  Eigldeen.        25 

A  primitive  group  of  degree  18  and  order  18.17.4  =  2^31 17  contains  among 
its  substitutions  476  of  degree  18,  459  of  degree  16,  288  of  degree  17.  According 
to  Sylow's  theorem  it  contains  either  1,  3,  9,  17,  51,  or  153  conjugate  subgroups 
of  order  2^.  Now  the  group  leaving  one  element  unchanged  contains  17  conju- 
gate subgroups  of  degree  16  and  order  4;  therefore  the  group  of  degree  18  contains 
153  distinct  conjugate  subgroups  of  order  4;  therefore  it  contains  153  conjugate  sub- 
groups of  order  2^.     Each  of  these  is  contained  self-conjugately  in  no  larger  group. 

The  number  of  systems  of  intransitivity  in  any  one  is  got  from  the  following 
equation,  where  x  denotes  the  number  of  substitutions  of  degree  18  and  a  the 
number  of  systems: 

18cc  +  16  (7  —  x)=:  8(18  —  a),  where  a  :^  1,  x-<  8. 

There  are  two  sets  of  solutions,  either  x-  =  0,  a  =  4,  or  a;  =  4,  a  =  3. 
The  group  of  degree  17  is  generated  by, 

where  t  and  its  powers  form  a  self- conjugate  subgroup  of  the  group  of  order  2^  and 
degree  18  that  is  now  under  discussion.  It  is  impossible  to  so  connect  the  systems 
and  introduce  the  remaining  elements  that  the  first  solution  may  give  the  group 
of  order  2^  Making  use  of  the  second  solution  we  have  only  to  combine  with 
the  group  generated  by  ^  a  substitution  of  degree  18  that  connects  the  two  remain- 
ing elements  by  a  transposition,  and  unites  the  cycles  of  ^  in  pairs.  The  153 
groups  of  order  2^  give  in  this  way  153.4=  612  distinct  substitutions  of  degree 
18,  while  there  are  only  476  in  the  group.  This  group  of  degree  18  does  not  exist. 
If  there  is  a  primitive  group  of  order  18  .  17.  2^  =  2^  3l  17,  it  contains  288 
substitutions  of  degree  17,  1071  of  degree  16,  1088  of  degree  18.  The  group  of 
degree  17  which  is  generated  by 

s  -=i  a^a^a-ia^a^a^a^a^aQaiifi-^ya^Mi2,ciiiai^a^^a^^ 
and  t  =  a^iCt^Qai/^ai^ayia^arflz .  ajjUi^a^aycft^^a^ai^a^ , 

contains  17  conjugate  subgroups  of  degree  16  and  order  8;  therefore  in  the 
group  of  degree  18  there  are  153  such  conjugate  subgroups,  and  each  of  these 
is  self-conjugate  in  a  group  of  order  2^  and  degree  18.  Denoting  by  a  the 
number  of  systems  of  intransitivity  of  this  group  of  order  2*,  and  letting  x  denote  the 
number  of  substitutions  of  degree  18  contained  in  the  group,  we  have  the  equation 
18a:  -h  16  (15  — ic)=  16  (18  — a),  where  a  ^l.  There  are  two  solutions,  x=  8, 
a=:2;a;=0,  a=3.     The  first  solution  would  involve  a  larger  number  of  substi- 


26  Martin:    On  the  Imprimitive  Substitution   Groups  of  Degree 

tutions  of  degree  18  than  are  actually  present  in  the  group  under  consideration. 
The  second  solution  shows  that  the  group  must  contain  153  conjugate  subgroups  of 
order  2^  and  degree  18  consisting  of  substitutions  of  class  16  only,  and  involving 
three  systems  of  intransitivity.  A  substitution  must  therefore  be  combined  with< 
that  transforms  t  into  one  of  its  powers,  and  has  its  head  in  the  group  generated 
by  t]  moreover,  this  substitution  must  have  as  one  of  its  cycles  the  transposition 
(ajajg),  and  must  have  systems  of  intransitivity  apart  from  this  cycle  consistent 
with  the  systems  of  t.  Such  a  substitution  is  a'=-a.^aiQ .  a^  «i4 .  ci^cLy, .  a^ni^  ■  Oo^is  •  «7«ii. 
a^ai^.ttiaiQ.     The  required  group  is  therefore  \s,  t,  a\. 

It  is  not  necessary  to  prove  that  these  three  substitutions  give  a  group  of 
the  required  order,  as  such  a  group  would  be  necessarily  doubly  transitive,  and 
it  is  known  that  there  is  a  doubly  transitive  group  of  degree  18  and  of  the 
required  order.  B}'  the  mode  of  construction  of  the  substitutions,  it  is  evident 
that  there  is  only  the  one  type  of  group  of  this  degree  and  order. 

Any  primitive  group  of  degree  18  and  order  18  .  17.  16  contains  2312  sub- 
stitutions of  degree  18,  288  of  degree  17,  2295  of  degree  16.  The  group  of 
degree  17  contains  17  conjugate  cyclical  subgroups  of  degree  16  and  order  16, 
therefore,  the  group  of  degree  18  contains  153  subgroups  of  order  16,  each  of 
which  is  self-conjugate  in  one  of  153  conjugate  subgroups  of  order  32.  Giving 
a  andx  the  usual  meanings,  we  find  that  the  group  of  order  32  involves  the  equa- 
tion 18a:  +  16  (31  — x)=  32(18  —  a),  where  a  =f=  1.  The  only  solution  is  a  =  2, 
x  =  8;  therefore,  the  group  of  degree  12  and  order  32  must  be  intransitive 
with  two  systems  of  intransitivity,  and  must  contain  8  substitutions  of  degree 
18,  23  of  degree  16.  We  have  to  add,  therefore,  to  the  cyclical  group  of  degree 
16,  8  substitutions  of  degree  16  and  8  of  degree  18,  all  of  them  containing  as  one 
cycle  the  transposition  of  the  remaining  two  letters. 

The  group  of  degree  17  and  order  17.16  has  as  generators 

and  u  =  a.M^a^^a^^a^^aQa^^ayMi^al^a^a^a^a^^a^a^ . 

The  substitution  t  =  (i\U\%  •  n^a^ .  a.^aio  •  ^s'^ii  •  ^'e^'s  •  «i/^i6  •  «n«i3  •  «i2<^*i5  generates  with 
s  and  u  the  required  group  of  degree  18  and  order  17.16.18.  A  triply  transi- 
tive group  of  such  an  order  is  known  to  exist  (Burnside,  1.  c,  p.  158);  so  no 
further  proof  that  \s,  u,  t:\  is  a  group  is  necessary.  It  is  easy  to  see  that  the 
even  substitutions  of  the  group  just  found  form  the  simple  group  of  order 
18 .17.8. 


Fifteen  and  the  Primitive  Substitution  Groups  of  Degree  Eigliteen.        27 

The  three  remaining  transitive  groups  of  degree  17  each  contains  120  con- 
jugate subgroups  of  order  17.  They  are  of  orders  15.16.17,  15. 16. 17. 2, 
15.16.17.4  respectively. 

The  group  of  degree  18  and  order  15  .  16  .  17  .  18  would  necessarily  contain 
816  conjugate  subgroups  of  order  5.  Each  is  self- conjugate  in  a  group  of  order 
90  connecting  the  remaining  three  elements  transitively.  This  group  is  intran- 
sitive with  two  transitive  constituents,  one  of  degree  15  and  order  90,  the  other 
of  degree  3.  The  first,  however,  is  non-existent,  therefore,  the  group  of  degree 
18  is  non-existent. 

The  two  remaining  groups  also,  if  they  can  generate  primitive  groups  of 
degree  18,  would  generate  groups  that  each  contain  816  conjugate  subgroups  of 
order  5.  In  the  one  case,  we  should  have  to  make  use  of  an  intransitive  group 
containing  as  a  transitive  constituent  a  group  of  degree  15  and  order  180,  in  the 
other,  the  transitive  constituent  would  enter  as  a  group  of  degree  15  and  order 
360.  Both  of  these  groups  are  non-existent;  therefore,  the  three  groups  of 
degree  17,  at  present  under  discussion,  furnish  us  with  no  new  groups  of 
degree  18. 

As  the  case  now  stands,  the  conclusion  arrived  at  may  be  summed  up  as 
follows : 

There  are  no  simply  transitive  primitive  groups  of  degree  18,  and  in  addi- 
tion to  the  symmetric  and  alternating  groups,  there  are  only  two  multiply  transi- 
tive groups  of  this  degree,  viz.,  the  two  given  by 

^  («2«io«i4«i6<^i7«9«6«3  •  «4«n<^6«i2«i5«8«i3«7) .  I  of  ordcr  2448, 

I  (ag^io  •  otgaii .  a^ay, .  a^a^^ .  a^a^^ .  a^a^^ .  a^a^^ .  a^a^^) ,  J 

r  (aia2«3«4«5«6«7«8^9«10«n«13«13«fl4«15«16«17)  »  "j 

\  («3«4«io«n^i4<^6«i6«i2<^i7«i5«9«8«5«i3<^3«7)»  }- of  Order  48 96 . 

The  second  of  these  is  triply  transitive,  and  contains  the  first,  which  is 
doubly  transitive  and  simple,  as  a  self-conjugate  subgroup. 

The  works  consulted  in  the  preparation  of  this  paper  have  included,  in  addi- 
tion to  the  standard  works  on  the  subject  by  Jordan,  Serret,  Netto,  and  Burn- 
side,  the  following  papers  : 

Askwith,  "  On  Possible  Groups  of  Substitutions  that  can  be  formed  with 
3,  4,  5,  6,  7  Letters  Respectively."  Quar.  Jour.  Math.,  v.  XXIV  (1890),  pp. 
111-167. 


28       Martin  :    On  the  Imprimitive  Substitution   Groups  of  Degree^  etc. 

"  On  Groups  of  Substitutions  that  can  be  formed  with  Eight  Letters."  Quar. 
Jour.  Math.,  v.  XXIV  (1890),  pp.  263-331. 

"On  Groups  of  Substitutions  that  can  be  formed  with  Nine  Letters,"  Quar. 
Jour.  Math.,  v.  XXVI  (1892),  pp.  79-128. 

Cayley,  "  On  Substitution  Groups  for  Two,  Three,  Four,  Five,  Six,  Seven, 
and  Eight  Letters."     Quar.  Jour.  Math.,  v.  XXV  (1891),  pp.  71-88,  137-155. 

Cole,  "  List  of  the  Substitution  Groups  of  Nine  Letters."  Quar.  Jour.  Math., 
V.  XXVI  (1892),  pp.  372-388. 

"The  Transitive  Substitution  Groups  of  Nine  Letters."  I3ull.  New  York 
Math.  Soc,  V.  II  (1893),  pp.  250-258. 

"List  of  the  Transitive  Substitution  Groups  of  Ten  and  of  Eleven  Letters." 
Quar.  Jour.  Math.,  v.  XXVII  (1894),  pp.  39-50. 

"Note  on  the  Substitution  Groups  of  Six,  Seven,  and  Eight  Letters."  Bull. 
New  York  Math.  Soc,  v.  II  (1893),  pp.  184-190. 

Miller,  "  Intransitive  Substitution  Groups  of  Ten  Letters."  Quar.  Jour. 
Math.,  V.  XXVII  (1894),  pp.  99-118. 

"List  of  Transitive  Substitution  Groups  of  Degree  Twelve."  Quar.  Jour. 
Math.,  V.  XXVIII  (1896),  pp.  193-231. 

"Note  on  the  Transitive  Substitution  Groups  of  Degree  Twelve."  Bull. 
Amer.  Math.  Soc,  v.  I  (1894-1895),  pp.  255-258. 

"On  the  Transitive  Substitution  Groups  of  Degrees  Thirteen  and  Fourteen." 
Quar.  Jour.  Math.,  v.  XXIX  (1898),  pp.  224-249. 

"  On  the  Primitive  Substitution  Groups  of  Degree  Fifteen."  Proc  Lon. 
Math.  Soc,  V.  XXVIII  (1896-1897),  pp.  533-544. 

"On  the  Primitive  Substitution  Groups  of  Degree  Sixteen."  Amer.  Jour. 
Math.,  V.  XX  (1898),  pp.  229-241. 

"On  the  Transitive  Substitution  Groups  of  Degree  Seventeen."  Quar. 
Jour.  Math.,  v.  XXXI  (1899),  pp.  49-57. 

"A  Simple  Proof  a  Fundamental  Theorem  of  Substitution  Groups,  and 
Several  Applications  of  the  Theorem."  Bull.  Amer.  Math.  Soc,  v.  II  (1895- 
1896),  pp.  75-77. 

"  On  the  Limit  of  Transitivity  of  the  Multiply  Transitive  Substitution  Groups 
that  do  not  Contain  the  Alternating  Group."  Bull.  Amer.  Math.  Soc,  v.  IV 
(1897-1898),  pp.  140-143. 

Jordan,  "  Sur  la  classification  des  groupes  primitifs,"  Comptes  Rendus,  t. 
LXXIII  (1871  j,  pp.  853-857. 

"  Sur  Venumeration  des  groupes  primitifs  pour  les  dix-sept  premiers  degres." 
Comptes  Rendus,  t.  LXXV  (1872),  pp.  1754-1757. 

Philadelphia,  January  1901. 


LIFE. 

I  was  born  in  Elizabeth,  New  Jersey,  December  30,  1869.  In  1890,  I 
entered  Bryn  Mawr  College,  selecting,  as  my  major  studies,  Mathematics  and 
Latin.  In  1894,  I  received  the  degree  of  A.  B.  from  this  college.  The  first 
semester  of  the  following  year,  1894-1895,  was  spent  at  Bryn  Mawr  College  as 
a  graduate  student  in  Mathematics  and  Physics,  the  second  semester,  in  teaching 
at  a  preparatory  school.  During  the  year  1895-1896  I  held  the  Fellowship  in 
Mathematics  in  Bryn  Mawr  College,  remaining  there  the  following  year,  1896- 
1897,  as  a  graduate  student.  During  the  year  1897-1898  I  held  the  Mary  E. 
Garrett  European  Fellowship  from  Bryn  Mawr  College,  and  spent  the  entire 
year  at  the  University  of  Gottingen,  where  I  attended  the  lectures  of  Professors 
Klein  and  Hilbert.  I  then  returned  to  Bryn  Mawr  College,  where  I  was  Fellow 
by  Courtesy  in  Mathematics  during  the  year  1898-1899.  In  the  spring  of  1899  I 
passed  the  examination  at  Bryn  Mawr  College  for  the  degree  of  Doctor  of 
Philosophy.  My  major  subject  was  Mathematics,  pursued  under  the  direction 
of  Professors  Scott  and  Harkness,  while  my  double  minor  was  Physics,  pursued 
under  the  direction  of  Professor  Mackenzie. 

My  gratitude  is  due  to  all  the  Professors  under  whom  I  have  studied,  and 
especially  to  Professor  Harkness,  under  whose  direction  this  paper  was  prepared. 


21642.'? 


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